## Thursday, 1 October 2020

### The Decimal point and Place Value

My daily twitter browsing showed me the recurring argument about "moving the decimal point" in relation to multiplication or division by 10, 100 , 1000 etc:

The thread is worth a read (you can get to it if you click the image). I read it, and some of the quote tweets around it, and it got me thinking about what I think are two possible misconceptions around the decimal point and place value that seem to exist in the minds of teachers of maths. I am sure some will disagree with one or both of these, and may think that these are misconceptions in my mind instead. But anyway, here goes...

Misconception 1: That the decimal point is a fixed immutable point between the ones and the tenths

There are a number of people saying in that thread that the decimal point cannot move, that it must stay between the ones and the tenths. This for me seems false. The job of the decimal point is to separate or mark the transition between whole values of our unit, and values that represent part of that unit. Of course, in most cases our unit of counting would be ones and the decimal point therefore marks the transition from ones to part of one. But that isn't always the case. Consider for example:

2.6 million

In this, the value to the left of the decimal point does not represent two ones, it represents two millions. The unit we have chosen to count in is millions, and so the decimal point separates the whole millions from the parts of a million. This has obvious parallels to something most mathematicians will be familiar with converting between standard form and our "ordinary" decimal number system:

320000 = 3.2 × 105

In converting to standard form we are literally changing our counting unit from the ones to which column has the highest value in the number we are working with. This means that the decimal point does move, it moves to separate whole values of our new counting unit (in the above example 105) from parts of this counting unit. In converting from standard form we do the reverse; we change our counting unit from the largest valued column back to the ones column, and the decimal point moves concurrently. One could even make the argument that converting units of measure could be viewed in the same way:

3.25 metres = 325 cm

Have we multiplied by 100 to go from left to right here? The physical distance hasn't changed? Would it make more sense to consider that the decimal point has moved due to our change of unit choice from metres to centimetres, and so what was separating whole metres from part metres, is now separating whole centimetres from part centimetres.

Misconception 2: That the decimal point moves when we multiply or divide by a power of 10

Despite what I have written above, and some compelling arguments within the thread, I still come down on the side that it is wrong to teach pupils that the decimal point moves when you multiply or divide by a power of 10. This is not just because of its tendency to be used a trick for teaching without understanding, but more because conceptually it actually doesn't fit with what is happening when we multiply or divide. If we accept that the decimal point moves when we decide to change our unit (perhaps a big if for some), then the decimal point cannot move when we multiply or divide by a power of 10. Consider:

3.2 × 100 = 320

There is no change of counting unit in this situation. In all three numbers, the counting unit is ones, and the decimal point separates the ones for the parts of one. What has changed is the physical size of the numbers that each of the digits represents in the number 3.2. The 3 has become 100 times bigger to become 300, and the 0.2 has become 100 times bigger to become 20. This is synonymous with moving the digits up the place value columns, and definitely not the decimal point down - or even translating the column headings down. Although I can see the argument that says "we can consider 3.2 × 100 as having 3.2 hundreds, and what we are doing is converting that back to a number of ones" I can't make that fit in my own mind with the importance of making sure pupils recognise and appreciate the multiplicative relationships between the place value columns. Even if our learners are completely secure with this, I can't see why we would then use a unitising approach to multiplication to model multiplicative calculations with different powers of 10 - except maybe if we had reached the point where pupils were so secure in this that we were opening them up to another way of making sense of such calculations and perhaps attempting to highlight that moving the decimal point is akin to the equivalent multiplication or division.

In summary, the decimal point definitely can move, but probably shouldn't if we are teaching multiplication or division by powers of 10 unless we are taking a unitising approach to this sort of calculation.